We study the partial regularity of weak solutions to the 2-dimensional Landau- Lifshitz equations coupled with time dependent Maxwell equations by Ginzburg-Landau type approximation. Outside an energy concentration set of locally finite 2-dimensional parabolic Hausdorff measure, we prove the uniform local C∞ bounds for the approaching solutions and then extract a subsequence converging to a global weak solution of the Landau-Lifshitz-Maxwell equations which are smooth away from finitely many points.
In this paper, the authors discuss the vortex structure of an anisotropic Ginzburg-Landau model for superconducting thin film proposed by Du. We obtain the estimate for the lower critical magnetic field Hc1 which is the first critical value of hex corresponding to the first phase transition in which vortices appear in the superconductor. We also find local minimizers of the anisotropic superconducting thin film with a large parameter k, and for the applied magnetic field near the critical field we discuss the asymptotic behavior of the local minimizers.
In this paper, we discuss the vortex structure of the superconducting thin films placed in a magnetic field. We show that the global minimizer of the functional modelling the superconducting thin films has a bounded number of vortices when the applied magnetic field hex 〈 Hc1 ＋ K log ｜ logε｜ where Hc1 is the lower critical field of the film obtained by Ding and Du in SIAM J. Math. Anal., 2002. The locations of the vortices are also given.
We consider the partial regularity of weak solutions to the weighted Landau-Lifshitz flow on a 2-dimensional bounded smooth domain by Ginzburg-Landau type approximation. Under the energy smallness condition, we prove the uniform local C^∞ bounds for the approaching solutions. This shows that the approximating solutions are locally uniformly bounded in C^∞（Reg（{uε}）∩（Ω^-×R^＋）） which guarantee the smooth convergence in these points. Energy estimates for the approximating equations are used to prove that the singularity set has locally finite two-dimensional parabolic Hausdorff measure and has at most finite points at each fixed time. From the uniform boundedness of approximating solutions in C^∞（Reg（{uε}）∩（Ω^-×R^＋））, we then extract a subsequence converging to a global weak solution to the weighted Landau-Lifshitz flow which is in fact regular away from finitely many points.
In this paper, we study the initiial-boundary value problem of one class of nonlinear Schrodinger equations described in molecular crystals. Furthermore, the existence of the global solution is obtained by means of interpolation inequality and a priori estimation.
In this article, using coordinate transformation and Gronwall inequality, we study the vortex motion law of the anisotropic Cinzburg-Landau equation in a smooth bounded domain Ω （R^2,that is ,Эtuε=j,k=1∑2（ajkЭxkuε）xj＋ε^2^-b（x）（1-｜uε｜^2）uε,x∈Ω,and conclude that each vortex,bj（t）（j=1,2,…,N）satisfies dt^-dbj（t）=-（a（bj（t））^-a1k（bj（t））Эxka（bj（t））,a（aj（t））^-a2k（bj（t））Эxka（bj（t）））,where a（x）=√a11a22-a12^2. We prove that all the vortices are pinned together to the critical points of a（x）. Furthermore, we prove that these critical points can not be the maximum points.
